Changes between Version 25 and Version 26 of PolynomialExpansion


Ignore:
Timestamp:
01/26/16 14:59:20 (10 years ago)
Author:
sili
Comment:

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  • PolynomialExpansion

    v25 v26  
    11
    2 A = {{X0, X1}, {X1, X2}},  b = X3[2]
     2A = {{a,,0,,, a,,1,,}, {a,,1,,, a,,2,,}},  b = {b,,0,,, b,,1,,}
    33   
     4
    45Step 1: r = b - Ax
    5   r[0] = X3[0]
     6  r[0] = b,,0,,
    67
    7   r[1] = X3[1]
     8  r[1] = b,,1,,
    89
    910Step 2: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
    10   p[i]*(A[i][j]*p[j]) = X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^
     11  p[i]*(A[i][j]*p[j]) = a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^
    1112
    12   alpha = (X3[0]^2^+X3[1]^2^) / ((X3[1]^2^)*X2+2*(X3[1]*X3[0]*X1)+(X3[0]^2^)*X0)
     13  alpha = (b,,0,,^2^+b,,1,,^2^) / ((b,,1,,^2^)*a,,2,,+2*(b,,1,,*b,,0,,*a,,1,,)+(b,,0,,^2^)*a,,0,,)
    1314
    1415Step 3: r[i] = r - alpha * A[i][j] * p[j]
    15   r[0] = (-X3[1]*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
     16  r[0] = (-b,,1,,*(-a,,1,,*b,,0,,^2^ + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,^2^)) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)
    1617
    17   r[1] = (X3[0]*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
     18  r[1] = (b,,0,,*(-a,,1,,*b,,0,,^2^ + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,^2^)) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)
    1819
    1920Step 4: x[i] = x + alpha*p[i]
    20   X[0] = (X3[0]*(X3[0]^2^ + X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
     21  X[0] = (b,,0,,*(b,,0,,^2^ + b,,1,,^2^)) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)
    2122
    22   X[1] = (X3[1]*(X3[0]^2^ + X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)
     23  X[1] = (b,,1,,*(b,,0,,^2^ + b,,1,,^2^)) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)
    2324
    2425Step 5: beta = rsnew / rsold = (rk[i]*rk[i]) / (r[i]*r[i])
    25   rsnew = ((X3[0]^2^ + X3[1]^2^)*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)^2^) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
     26  rsnew = ((b,,0,,^2^ + b,,1,,^2^)*(-a,,1,,*b,,0,,^2^ + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,^2^)^2^) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)^2^
    2627
    27   beta = (X1*X3[0]^2^ - X0*X3[0]*X3[1] + X2*X3[0]*X3[1] - X1*X3[1]^2^)^2^ / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
     28  beta = (a,,1,,*b,,0,,^2^ - a,,0,,*b,,0,,*b,,1,, + a,,2,,*b,,0,,*b,,1,, - a,,1,,*b,,1,,^2^)^2^ / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)^2^
    2829
    2930Step 6: p[i] = rk[i] +beta * p
    30   p[0] = (-1)*((X1*X3[0] + X2*X3[1])*(X3[0]^2^ + X3[1]^2^)*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
     31  p[0] = (-1)*((a,,1,,*b,,0,, + a,,2,,*b,,1,,)*(b,,0,,^2^ + b,,1,,^2^)*(-a,,1,,*b,,0,,^2^ + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,^2^)) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)^2^
    3132
    32   p[1] = ((X0*X3[0] + X1*X3[1])*(X3[0]^2^ + X3[1]^2^)*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^2^
     33  p[1] = ((a,,0,,*b,,0,, + a,,1,,*b,,1,,)*(b,,0,,^2^ + b,,1,,^2^)*(-a,,1,,*b,,0,,^2^ + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,^2^)) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)^2^
    3334
    3435Step 7: alpha = (r[i]*r[i]) / (p[i]*A[i][j]*p[j])
    35   p[i]*(A[i][j]*p[j]) = ((-X1^2^ + X0*X2)*(X3[0]^2^ + X3[1]^2^)^2^*(-X1*X3[0]^2^ + X0*X3[0]*X3[1] - X2*X3[0]*X3[1] + X1*X3[1]^2^)^2^) / (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^)^3^
     36  p[i]*(A[i][j]*p[j]) = ((-a,,1,,^2^ + a,,0,,*a,,2,,)*(b,,0,,^2^ + b,,1,,^2^)^2^*(-a,,1,,*b,,0,,^2^ + a,,0,,*b,,0,,*b,,1,, - a,,2,,*b,,0,,*b,,1,, + a,,1,,*b,,1,,^2^)^2^) / (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^)^3^
    3637
    37   alpha = (X0*X3[0]^2^ + 2*X1*X3[0]*X3[1] + X2*X3[1]^2^) / ((-X1^2^ + X0*X2) (X3[0]^2^ + X3[1]^2^))
     38  alpha = (a,,0,,*b,,0,,^2^ + 2*a,,1,,*b,,0,,*b,,1,, + a,,2,,*b,,1,,^2^) / ((-a,,1,,^2^ + a,,0,,*a,,2,,) (b,,0,,^2^ + b,,1,,^2^))
    3839
    3940Step 8: r[i] = r - alpha * A[i][j] * p[j]
    … …  
    4344
    4445Step 9: x[i] = x + alpha*p[i]
    45   X[0] = (X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^)
     46  X[0] = (a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,^2^)
    4647
    47   X[1] = (-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^)
     48  X[1] = (-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,^2^)
    4849
    4950assertion: 
    50   bncg[0] = A[0][0]*X[0] + A[0][1]*x[1] = X0*(X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^) + X1*(-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^) = X3[0]*(X0*X2 - X1^2^) / (X0*X2 - X1^2^) = X3[0]
     51  bncg[0] = A[0][0]*X[0] + A[0][1]*x[1] = a,,0,,*(a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,^2^) + a,,1,,*(-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,^2^) = b,,0,,*(a,,0,,*a,,2,, - a,,1,,^2^) / (a,,0,,*a,,2,, - a,,1,,^2^) = b,,0,,
    5152
    52   bncg[1] = A[1][0]*X[0] + A[1][1]*x[1] = X1*(X2*X3[0] - X1*X3[1]) / (X0*X2 - X1^2^) + X2*(-X1*X3[0] + X0*X3[1]) / (X0*X2 - X1^2^) = X3[1]*(X0*X2 - X1^2^) / (X0*X2 - X1^2^) = X3[1]
     53  bncg[1] = A[1][0]*X[0] + A[1][1]*x[1] = a,,1,,*(a,,2,,*b,,0,, - a,,1,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,^2^) + a,,2,,*(-a,,1,,*b,,0,, + a,,0,,*b,,1,,) / (a,,0,,*a,,2,, - a,,1,,^2^) = b,,1,,*(a,,0,,*a,,2,, - a,,1,,^2^) / (a,,0,,*a,,2,, - a,,1,,^2^) = b,,1,,
    5354
    54   b[0] = X3[0]
     55  b[0] = b,,0,,
    5556
    56   b[1] = X3[1]
     57  b[1] = b,,1,,
    5758
    5859END